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Full text of "Effect of Compliant Walls on Secondary Instabilities in Boundary-Layer Transition"

Effect of Compliant Walls on Secondary Instabilities in 
Boundary-Layer Transition 

Ronald D. Joslinj 
NASA Langley Research Center, Hampton, Virginia 23681 

and 

Philip J. Morris J 
Pennsylvania State University, University Park, Pennsylvania 16802 

Abstract 

For aerodynamic and hydrodynamic vehicles, it is highly desirable to reduce drag and noise levels. A reduction 
in drag leads to fuel savings. In particular for submersible vehicles, a decrease in noise levels inhibits detection. 
A suggested means to obtain these reduction goals is by delaying the transition from laminar to turbulent flow 
in external boundary layers. For hydrodynamic applications, a passive device which shows promise for transition 
delays is the compliant coating. In previous studies with a simple mechanical model representing the compliant wall, 
coatings were found that provided transition delays as predicted from the semi-empirical e" method. Those studies 
were concerned with the linear stage of transition where the instability of concern is referred to as the primary 
instability. For the flat-plate boundary layer, the Tollmien-Schlichting (TS) wave is the primary instability. In one 
of those studies, it was shown that three-dimensional (3-D) primary instabilities, or oblique waves, could dominate 
transition over the coatings considered. From the primary instability, the stretching and tilting of vorticity in the 
shear flow leads to a secondary instability mechanism. This has been theoretical described by Herbert based on 
Floquet theory. In the present study, Herbert's theory is used to predict the development of secondary instabilities 
over isotropic and non-isotropic compliant walls. Since oblique waves may be dominant over compliant walls, a 
secondary theory extention is made to allow for these 3-D primary instabilities. The effect of variations in primary 
amplitude, spanwise wavenumber, and Reynolds number on the secondary instabilities are examined. As in the 
rigid wall case, over compliant walls the subharmonic mode of secondary instability dominates for low-amplitude 
primary disturbances. Both isotropic and non-isotropic compliant walls lead to reduced secondary growth rates 
compared to the rigid wall results. For high frequencies, the non-isotropic wall suppresses the amplification of the 
secondary instabilities, while instabilities over the isotropic wall may grow with an explosive rate similar to the rigid 
wall results. For the more important lower frequencies, both isotropic and non-isotropic compliant walls suppress 
the amplification of secondary instabilities compared to the rigid wall results. The twofold major discovery and 
demonstration of the present investigation are: (1) the use of passive devices, such as compliant walls, can lead to 
significant reductions in the secondary instability growth rates and amplification; (2) suppressing the primary growth 
rates and subsequent amplification enable delays in the growth of the explosive secondary instability mechanism. 



1. Introduction 

Research involving fiow over flexible walls was 
started in the late-1950's by Kramer^'^. Experimentally, 
Kramer found significant drag reductions using rubber 
coatings over rigid walls. Investigators in the 1960's fo- 
cused on the task of experimentally duplicating and the- 
oretically explaining Kramer's results. The majority of 
these studies failed to produce any comparable results; 
yet, the theoretical results laid the foundation for all fu- 
ture studies involving fiexible walls. Interest turned to- 
ward the use of compliant walls for turbulent drag reduc- 
tion. In the 1970's NASA^ and in the 1980's the Office of 
Naval Research'* sponsored investigations involving the 
use of compliant walls for the turbulent problem. Al- 
thought most of the results from this era were either in- 
conclusive or unsatisfactory, the contributions, together 
with earlier results, have acted as stepping stones to the 
understanding of the physically 



complex fiuid/wall interaction phenomena. A compre- 
hensive review of the pioneering studies was given by 
Bushnell, Hefner and Ash®, in particular for the tur- 
bulent fiow problem. More recent reviews were given 
by Riley, Gad-el-Hak and Metcalfe®, Gad-el-Hak^'®, and 
Carpenter^. 

Motivation for the present investigation is partially 
derived from the following favorable theoretical and ex- 
perimental results. In the early 1980's, Carpenter and 
Garrad**^'** showed theoretically that Kramer-type sur- 
faces could lead to potential delays in transition. Fur- 
ther, they indicated deficiencies in previous investiga- 
tions which may have prevented their achieving results 
comparable to Kramer's. Only recently, experiments 
performed by Willis*^ and Caster*^ showed favorable re- 
sults using compliant walls. As outlined in the above 
mentioned reviews, a number of investigations in the 
past ten years have been conducted involving flexible 
walls. A main emphasis of these studies was to under- 



stand the physical mechanisms involved in the fluid/wall 
interaction of transitional and turbulent flows. Most of 
these studies focused on the 2-D instability problem, ex- 
cept Yeo^'* who showed that a lower critical Reynolds 
number existed for the isotropic compliant wall for 3-D 
instability waves. Carpenter and Morris^® and Joslin, 
Morris and Carpenter^® have shown that 3-D Tollmien- 
Schlichting waves can have greater growth rates over 
compliant walls than 2-D waves. However, they showed 
that, even though 3-D waves may be dominant, transi- 
tion delays are still obtainable through the use of com- 
pliant walls. They considered a compliant wall model 
used by Grosskreutz^^ for his turbulent boundary-layer 
experiments. 

In this paper, the growth rates and ampliflcation 
of secondary instabilities over compliant walls are pre- 
dicted and compared to the rigid wall results. A the- 
ory for secondary instabilities is used which is based on 
Floquet Theory and was developed by Herbert^®. This 
theory agrees remarkably well with experiments, in par- 
ticular those of Klebanoff, Tidstrom and Sargent^^ for 
peak-valley splitting (fundamental) and Kachanov and 
Levchenko^*^ for peak- valley alignment (subharmonic). 
The secondary instability theory is extended to allow for 
3-D primary instabilities, which are dominant over the 
compliant walls considered. Appropriate boundary con- 
ditions for the Grosskreutz^^ wall model are derived for 
the secondary analysis. 

In the next section, the primary instability problem 
is outlined. In section three, the secondary instability 
problem is discussed. The numerical methods, results, 
and a summary follow in the remaining sections. 

2. Primary Wave Model 

Results for 2-D and 3-D primary instabilities over 
compliant and rigid walls have been presented by 
Joslin^"'^, Morris and Carpenter"'^®'"'^®. Since the secondary 
instability theory is based on and includes a primary in- 
stability analysis, brief derivations of the dynamic equa- 
tions and boundary conditions for the 3-D primary insta- 
bility problem are included below. The disturbances are 
represented as travelling waves which may grow or decay 
as they propagate. Nonlinear coupling is ignored so that 
individual components of the frequency spectrum may 
be studied. Additionally, the quasi-parallel assumption 
is made. 

Consider an incompressible laminar, boundary-layer 
flow over a smooth flat wall. The Navier-Stokes equa- 
tions govern the flow. The Blasius proflle is used to 
represent the mean flow. 

A small-amplitude disturbance is introduced into 
the laminar flow. A normal mode representation is given 
as 



{v[,Q[}(x, y, z, t) ={vi,Qi}(y) exp[i(a;a cos (f) 

+za sin (f) — ujt)] + c.c. (1) 

where vi and ili are the complex eigenfunctions of nor- 
mal velocity and vorticity, respectively. To obtain a real 
solution, complex conjugate solutions denoted by c.c. are 
required, a is the wavenumber, w is the frequency, and (f) 
is the wave angle. In general, a and w are complex lead- 
ing to an ambiguity in the system. For temporal analy- 
ses, a is a real specifled wavenumber and w is the com- 
plex eigenvalue. For spatial analyses, w is a real specifled 
frequency and a is the complex eigenvalue. For the com- 
pliant wall problem, Joslin, Morris and Carpenter^® have 
shown that the use of eqn. (1) leads to an overestimation 
of the growth of the wave as it propagates. The wave ac- 
tually propagates in a nearly streamwise direction which 
is in the direction of the group velocity, not normal to the 
wave fronts. In the present paper, the secondary insta- 
bilities are investigated using this simple representation 
of the primary instabilities. Since the present approach 
is conservative, it should exemplify the beneflts of using 
compliant walls as a means to obtain transition delays. 
Also, a major emphasis and motivation of the present 
study is to determine the behavior, or response, of the 
phenomena-namely secondary instabilities-to compliant 
walls. 

If the normal mode relation (1) is substituted into 
the linearized form of the Navier-Stokes equations, the 
following nondimensional system results: 



where 



v'{" + ai{y)v'{ + a2{y)vi = (2) 

ai(y) = — iRs{Uo{y)ci cos (j) — uj) — 2a 
a-ziy) = iRja^ {Uo{y)a cos (f) -Lu) 
+iRsa cos <f)Uo"{y) + a 

n'\ + a3(j/)fii + a4iy)vi = (3) 

asiy) =- a^ - iRs(Uo(y)a cos (f) - w) 
a-iiy) = - i Rs a sin (t)U ' o(y)- 



The equations are nondimensionalized using the 
freestream velocity Uoo, kinematic viscosity i>, and 
an appropriate length scale. Convenient lengths for 
the boundary-layer scale with the ^-Reynolds number, 
ijj, = Uoox/i'. These include a thickness, 6, where 
the Reynolds number is deflned Rs = Rx and a 
boundary-layer displacement thickness, 6* , where Rs- = 



where 



-I/O 

17207i?j; . Eqns. (2) and (3) are referred to as the Orr- 
Sommerfeld and Squire equations, respectively. The sys- 
tem requires six boundary conditions. Requiring that 
the disturbance fluctuations vanish at infinity supplies 
three: 



vi{y) , v,'{y) , fii(j/)^0 



y 



(4) 



The remaining boundary conditions are determined from 
the compliant wall model. 

The compliant wall model used for the present pa- 
per was introduced by Grosskreutz^^ in his experimental 
drag reduction studies with turbulent boundary layers. 
He suggested that the link between streamwise and nor- 
mal surface displacements would cause a negative pro- 
duction of turbulence near the wall. Although his results 
for the turbulent fiow were disappointing, the surface 
does react to the fiuid fiuctuations in transitional fiow in 
such a way as to reduce production of instability growth. 
Carpenter and Morris^^ have shown by an energy analy- 
sis how the many competing energy-transfer mechanisms 
are infiuenced by the compliant wall presence. Of note 
is the reduced energy production by the Reynolds stress 
which may cause the reduced growth rates. Further, 
Joslin, Morris and Carpenter^® predicted that transition 
delays of 4-10 times the rigid wall transition Reynolds 
number were achievable with this coating. So the model 
has been extended to allow for a secondary instability 
analysis. 

The mechanical model consists of a thin, elastic 
plate supported by hinged and sprung rigid members in- 
clined to the horizontal and facing upstream at an angle, 
6, when in equilibrium. A sketch of the mechanical wall 
model is shown in Fig. 1. The boundary conditions are 
obtained by enforcing a balance of forces in the stream- 
wise and spanwise directions and the continuity of fiuid 
and wall motion. These are given below in linearized 
form. 








Fig. 1 Mechanical model representing the 

Grosskreutz compliant coating. 



For small displacements of an element out of equi- 
librium, the mechanical surface can be thought to move 
in a direction perpendicular to the rigid swivel-arm. The 
horizontal and vertical displacements (Ci,rii) are linked 
to the angular displacement (66) by 



^1 = £,56' sin 6* and 



m 



£66 cos 6 



(5) 



where £ is the length of the rigid-arm member. Equations 
of motion for the element in the streamwise and spanwise 
directions may be obtained by a balance of the forces of 
the fiuid fiuctuations acting on the surface and the forces 
due to the wall motion. These equations are 






dt^ 



dx^ 



2Bx 



dx'^dz'^ 



+B, ^) cos2 6 + Ke m - E^h 



5^6 



{—P -\- Tyy) COS 6 + Tyx SlU 6 COS 6 



and 






KsCi-E,b 



d\i 



sin 6 cos 6 
(6) 

(7) 



where 



{Bx, Bz} 



{Ex,E,}b^ 

12(1 - V^V;,) 



id Br 



yBxB;, 



Ci is the spanwise surface displacement, pm and h are the 
plate density and thickness; (B^, B^z, B^) are the fiexu- 
ral rigidities of the plate in the streamwise, transverse, 
and spanwise directions; {Ex, E^) are the moduli of elas- 
ticity of the plate; Ke,Ks are the effective streamwise 
and spanwise spring stiffness; p is the pressure fiuctua- 
tion which is obtained from the fiuid momentum equa- 
tions; and Tyx, Tyy and Ty^ are the streamwise, normal, 
and spanwise viscous shear stress fiuctuations in the fiuid 
acting on the wall. 

The terms on the left hand side of eqn. (6) refer 
to mechanical forces and the terms on the right refer to 
fiuid motion forces due to viscous stress and pressure 
fiuctuations. For the case where the ribs are aligned 
at 6 = Q° , the wall becomes isotropic and reduces to the 
theoretical model studied by Carpenter and Garrad^*^'^^. 
Otherwise the wall is referred to as non-isotropic and the 
rib angle is determined by 6. 

The continuity of fiuid/wall motion is given in the 
streamwise, normal, and spanwise directions, respec- 
tively as 

56 
dt 
drji 
dt 

dt 



— — = Ml +r]iUo 



Vl 



Wl 



(8) 

(9) 

(10) 



where (ui,vi,wi) are the disturbance velocity compo- 
nents in the streamwise, normal and spanwise directions. 
For the Grosskreutz coating, Ks ^ oo is assumed, which 
from eqn. (7) would result in zero effective spanwise 
surface displacement. This implies from eqn. (10) that 
wi(0) = 0. Strictly speaking, if the assumption Ks -^ co 
is relaxed, the resulting instabilities have larger growth 
rates. This suggests that spanwise stiffeners are stabiliz- 
ing to a disturbed flow. So with the assumption enforced, 
a better coating for potential transition delays results. 
The surface displacement takes the same normal mode 
form as the primary wave given by eqn. (1). The normal 
modes are substituted into eqns. (6-10). The equations 
can be reduced to three equations in terms of the normal 
velocity and vorticity-'^®'^"'^. 

3. Secondary Instability Theory 

In this section, a secondary instability theory devel- 
oped by Herbert^® is extended to allow for 3-D primary 
instabilities. Additionally, boundary equations describ- 
ing the compliant walls are introduced for secondary in- 
stabilities. The flow is governed by the Navier-Stokes 
equations. Instantaneous velocity and pressure compo- 
nents are introduced and given as 

v(x,y,z,t) = V2(x,y,z,t) + Bv3(x,y,z,t) 

p{x, y, z, t) = p'zix, y, z, t) -^ Bpz{x, y, z, t) (11) 

where ps and v^ = (m3,i'3,W3) are the secondary dis- 
turbance pressure and velocity in the fixed laboratory 
reference frame (x,y,z); and p2 and V2 = (m2,^2,m'2) 
are the basic pressure and velocity given by, 

V2{x,y,z,t) = {Uo{y),0,0} + A{ui,vi,wi}{x,y,z,t) 
P2{x,y,z,t) = Api{x,y,z,t) (12) 

The basic fiow is given by the Blasius profile and eigen- 
functions of the primary wave. Assume locally that the 
primary wave is periodic in t and periodic in {x, z) with 
wavelength A^ = 2t: / a^ and efine a disturbance phase 
velocity which is 

Cr = {Cx = LOr I Ctr COS (/), 0, C^ = LO^ / Ctr sin (j)) . 

Then in a frame moving with the primary wave, 

vi{x, y, z) = vi(x, y, z) = vi(x + \x,y, z + X;,) (13) 

where (x, z) is the reference frame moving with the wave. 
With an appropriate normalization of primary eigen- 
functions {ui,vi,wi) the amplitude. A, directly mea- 
sures the maximum streamwise rms fiuctuation. This 
is given by 

max |«i(j/)p = |«i(j/™)P = 1/2 (14) 

0<y <oo 



The instantaneous velocities and pressure (11) are 
substituted into the Navier-Stokes equations which are 
linearized with respect to the secondary amplitude, B. 
The disturbance pressure is eliminated, resulting in the 
vorticity equations. 

-^V2 - 1-1 n3-{v2 ■ V)fi3 - (V3 ■ V)fi2 

Kg Ot i 

+ (^2 • V)V3 + (^3 • V)V2 = (15) 

with the continuity equation 

V -173 = (16) 

As with the primary problem, the equations are formed 
in terms of the normal velocity (^3) and vorticity {^3). 
The equations are found to take the form^^, 

[7^V2-^-(t^o-C.)f + C A]fi3 

^Rj ot ox OZ-' 



+ A{-(Fi-V)fi3-(i73-V)fii 



dUo dv3 
dy dz 

+ ( _|_ ^^^ _|_Q ^_|_ r _ ^^^ 

dz dx dy dx dz 

dvi du3 dvi dw3 ~i 
dz dy dx dy J 



(17) 



and, 



iir^ -7^-iUo-c.)T- + c,T-]v'v3■^ 



- Rs 



d_ 
dt 



d_ d_ 
' dx dz" 



dy"^ dx 



I dy dx'-^ dz'-^ dy'-' 

^d'^vi. d ,d'^vi d'^vi d'^vi „ 9^mi , d 

_l_ 2 —) 1- ( - -| - -\- 2 —) — 

dxdy dx dz'^ dx^ dy^ dxdy dy 

.d'^wi d'^wi d'^ui d'^vi . d ,dvi „9mi, 
_|_ ( i i _| i —\ 1_ ( — 1 _|_ 2 — -) 

dy'^ dx^ dxdz dydz dz dy dx 

d^ d^ d^ dvi d^ r)/9ui dwi d^ 



dz"^ dx"^ dy"^ dx dxdy dz dx dxdz 

dvi d"^ 1 ^ r_ A^v^t; ) ^ 2(^^ - ^^ -F ^^ 

dz dydz dx dz"^ dx"^ dy"^ 

d'^ui d ^.d'^wi d'^vi d dvi d^ d^ 

dxdy dx dxdy dxdz dz dx dx'-' dz'-' 

dy'-' dy dx dxdy dx dydz 

dvi 9^ 1 r 9 2 \ o/'^^"! "^^^i \ ^ 

dz dxdz dz dydz dxdz dx 



dvi , d^ 9^ , „9mi d^ 



dz dx'' dy'' dz dxdy 



]W3} 







(18) 



The disturbance quantities V3, Q3 and dv3/dy are re- 
quired to vanish far from the wall and at the wall for the 



rigid wall case. The compliant wall equations give the re- 
maining boundary conditions in the compliant case. Ad- 
ditionally, the primary amplitude, A, is a parameter in 
the equations and is assumed to be locally non-varying. 
As j4 ^ the Orr-Sommerfeld and Squire equations re- 
sult. For the case of interest where A^ 0, the primary 
eigenfunctions (mi, vi, wi) appear in the equations as co- 
efficients. 

To solve the secondary problem, a normal mode so- 
lution having the following form is assumed, 

V3{x,y,z,t) = e^'+'f^^' ""^ "^-^ "''' "^Wix, y, z) (19) 

where /3 = 2t: /X^, is a specified spanwise wavenumber 
and (T = (Tr -|- iiTi is a temporal eigenvalue or is spec- 
ified for spatial analyses. V(x,y,z) is a function that 
represents the class of secondary modes. Floquet theory 
suggests the form of solution for periodic systems. For 
the present problem, this may be written 



V(x, y, z) = eT(^ ''°'"f'+' """^ 'f''^V(x, y, z) 



(20) 



where 7 = 7^ + ^Ti is the characteristic exponent and 
V{x, y, z) is periodic in the {x, z) plane and may be rep- 
resented by a Fourier series. Thus the representation of 
the secondary instability for a 3-D basic fiow is. 



vs =e 



i7t-\-il3(z cos (f) — x sin (f)')-\-^(x cos (f)-\-z sin (/>) 



E ''-^yy^" 



l2)ar{x cos (j)-\-z sin (/>) 



(21) 



n = — 00 



This suggests a form of solution for the secondary distur- 
bance based on a coordinate system oriented at an angle 
4> with respect to the mean fiow and moving with the 
primary wave. If the coordinate system is aligned with 
the primary wave, or (/) = 0°, then the solution for the 
secondary disturbance would follow Herbert, Bertolotti 
and Santos^^ who considered a 2-D primary wave. 

If solutions given by eqn. (21) are substituted into 
eqns. (17) and (18), an infinite system of ordinary differ- 
ential equations result. The dynamic equations are de- 
termined by collecting terms in the governing equations 
with like exponentials. The system consists of two dis- 
tinct classes of solution because the even and odd modes 
decouple. Even modes correspond to the fundamental 
mode of secondary instability, and the odd modes are 
the subharmonic mode. Only a few terms of the Fourier 
series are retained since, as shown by Herbert, Bertolotti 
and Santos^^, this provides a sufficiently accurate ap- 
proximation for a 2-D disturbance. 

This form of solution indicates two complex quanti- 
ties, (T and 7, which leads to an ambiguity similar to 
that found with the Orr-Sommerfeld/Squire problem. 
There are four unknowns, <Tr,<^i,lr,li- Two can be de- 
termined while two must be chosen in some other way. 



In the present study, temporally-growing tuned modes 
are examined. The temporal growth rate is cr^, and <Ti 
can be interpreted as a shift in frequency. In this case, 
7^ = 7i = 0. If (Tj- = 0, then the secondary disturbance 
is travelling synchronously with the basic fiow. 

The boundary conditions for the secondary distur- 
bance are given as 



Vn,v',^n ^0 



y 



(22) 



along with the compliant wall boundary conditions. In 
the rigid wall case. 



,f^n=0 



at 



y 







(23) 



The analysis for the compliant boundary conditions for 
secondary instabilities follows the same route as was 
taken for the primary instabilities, except a number of 
additional terms arise due to the presence of the primary 
wave. 

The fiuid/wall motion must be continuous in each 
direction. In addition, the equations of force (6, 7) must 
balance in the streamwise and spanwise directions in the 
reference frame moving with the primary wave. Con- 
sistent with the fiuid equations, the amplitude of the 
primary wave is assumed to be locally non-varying. In 
deriving the final form of the wall equations, a signifi- 
cant difference between the primary and secondary form 
arises from the pressure contribution. The pressure for 
the secondary disturbance is determined from the mo- 
mentum equations which are complicated by primary 
coupling terms. 

The continuity of motion between the fiuid and solid 
is given by 



56 
dt 



-- «3 + mK + ^{(6 • v)«3 + (6 • v)«i} (24) 

^ = ^3 + a[{1, ■ V)vs + (6 • V)t;i } (25) 



^ = ws + A[{i,-V)ws + {i^-V)w,] (26) 



where 



-: „ ^ d d ^ d 

ox ay oz 



Equations (24-26) involve six unknowns for the velocity 
fiuctuation and surface displacement in a highly coupled 
system. As with the primary boundary conditions, it 
is possible to derive a set of equations which represent 
the surface motion in terms of the normal velocity and 
vorticity only. This is algebraically very tedious. A com- 
plete derivation is given by Joslin^"'^. Note that if A = 
in the secondary wall equations, the primary wall equa- 
tions result. This occurred with the fiuid equations as 
well. 



4. Numerical Methods of Solution 

The algebraic complexity of the dynamic equations 
for the secondary disturbance and the compliant wall 
equations requires that care be taken in applying any 
numerical technique. Because no theoretical or exper- 
imental data are available for the compliant problem, 
both shooting^® and spectraP^ approximations are used. 
Also, noting that Bertolotti^'* has shown for the rigid 
wall problem with a 2-D primary instability that, after 
the transformation from spatial to temporal, the solu- 
tions are in good agreement, a temporal analysis is pre- 
sented in this paper. 

For the spectral method, Chebyshev series are intro- 
duced to approximate each mode of the Fourier series. 
An algebraic transformation is used to change from the 
Chebyshev spectral domain [—1, 1] to the physical do- 
main. Due to the properties of the Chebyshev polyno- 
mial, the equations are recast in integral form. Cheby- 
shev polynomials are used to represent the basic flow in 
the series which are substituted into the integral equa- 
tions. For the basic flow, 35 polynomials provided suffi- 
cient resolution of the eigenfunctions. The series repre- 
senting the secondary instability requires 40 polynomi- 
als for sufficient convergence to the dominant eigenvalue. 
For the shooting method, beginning with the equations 
for the compliant wall, integrations of the disturbance 
equations across the boundary layer are performed us- 
ing a Runge-Kutta scheme. At the edge of the bound- 
ary layer, the numerical solution vectors are matched 
with the asymptotic solutions. A very accurate initial 
guess is found to be required for convergence using this 
method. To demonstrate the accuracy of the numerical 
techniques, a comparison for the rigid wall case is made 
with Herbert^s for Rs = 826.36, Fr = 83, /3 = 0.18, 
and A = 0.02. Herbert obtained the dominant mode 
a = 0.01184. In good agreement, the present spec- 
tral and shooting methods lead to cr = 0.011825 and 
a = 0.011839, respectively. 

5. Results 

For all of the results that follow, the freestream ve- 
locity is 20 m/s, the density is 1000 kg/m^, and the kine- 
matic viscosity is 1 x 10~® m^/s. The coatings considered 
consist of both isotropic and non-isotropic walls. Both 
walls were optimized at Rs- = 2240 for 2-D primary 
instabilities. The isotropic wall has properties 9 = 0°, 
h = 0.735mm, E^ = 1.385MN/m2, K = 0.354GN/m3 
and Pfn = 1000 kg/m^; and the non-isotropic wall has 
properties = 60°, h = 0.111mm, E^ = 0.509MN/m2, 
K = 0.059GN/m3 and pm = 1000 kg/m^. A Reynolds 
number of 2240 was chosen because, for a boundary layer 
over a rigid wall, the disturbance with the critical fre- 
quency (in the e" sense) reaches its maximum growth 
rate near this value of Reynolds number. Accordingly, 



this is a good choice of Reynolds number for optimizing 
the wall properties. In considering 3-D instabilities, the 
walls optimized for 2-D instabilities are used with the ad- 
dition of isotropic plates. The properties of an isotropic 
plate are direction independent; that is, E^ = E^,. Al- 
though complete details of the optimization process and 
philosophy are given by Carpenter and Morris^^, a recap 
follows. 

With a flexible wall present, other modes of instabil- 
ity arise. With changes in the compliant wall properties, 
stable, or marginally stable, fluid and wall modes can be- 
come unstable and dominant. The present wall proper- 
ties were varied to achieve an optimal specifled condition. 
This desired condition was to achieve a minimum growth 
rate for a dominant 2-D Tollmien-Schlichting instability 
while keeping other modes marginally stable. For the 
secondary analysis, these "optimal" compliant walls led 
to no additional unstable modes. However, this is not to 
say that additional growing modes may not appear for 
different wall properties. 

In this analysis, the primary wave amplitude {A) 
and the secondary instability spanwise wavenumber (/3) 
are parameters of the problem. Herbert^® showed in his 
boundary-layer studies for the 2-D primary wave over a 
rigid wall that as the amplitude increases the growth rate 
of the secondary instability increases. Also, as the span- 
wise wavenumber is varied, the temporal growth rate 
reaches a maximum for a particular wavenumber. Ad- 
ditionally, Herbert showed that the subharmonic mode 
reaches greater growth rates than the fundamental mode 
for low amplitude disturbances. These flndings were 
verifled by the direct numerical simulations of Spalart 
and Yang^^. Althought both the subharmonic and fun- 
damental modes over the compliant walls were exam- 
ined, emphasis is placed on the subharmonic mode, since 
as both theory and computations indicate, subharmonic 
disturbances are more unstable than fundamental distur- 
bances for small amplitudes. Limited fundamental dis- 
turbance results are included to verify that these modes 
do not become the dominant instability over compliant 
walls. 

Primary waves with frequencies that give maxi- 
mum disturbance growth rates are considered. For the 
isotropic wall, the maximum growth rate occurs at a fre- 
quency Lu = 0.065 (Fr =~ 29.0), where Fr = w/Rx 10'^. 
Figure 2 shows the growth rates of the subharmonic and 
fundamental disturbances as a function of the spanwise 
wavenumber for the rigid wall and isotropic compliant 
wall. As the flgure shows, growth rates over the com- 
pliant wall are reduced in comparison with the rigid 
wall results over the whole range of spanwise wavenum- 
bers. Additionally, the subharmonic disturbance has 
much larger growth rates than the fundamental distur- 
bance, as expected. Similar trends are found in the com- 



parison of non-isotropic and rigid wall results. In consid- 
ering reductions in the growth rates of the subharmonic 
mode as a result of compliant walls, the isotropic wall 
suppressed the maximum growth rate by 20%. For the 
maximum growth rate over the non-isotropic wall, which 
occurs at a difference frequency than the isotropic wall 
case, the non-isotropic wall led to a reduction of 17% 
compared to rigid wall results. So for a fixed Reynolds 
number and primary wave amplitude, both isotropic and 
non-isotropic compliant walls lead to reduced secondary 
instability growth rates compared to the rigid wall re- 
sults. 

.010 J- 



f 

i 




.IXID 

d 1 3 

S^UbIH WUHBUnbd , & 

Fig. 2 Growth rates of the secondary instabil- 
ities as a function of spanwise wavenuniber for 

Rs, = 2240, Fr ~ 29.0, and A = 0.01. subharmonic: 
— o— , rigid wall; • • o- • •, isotropic wall and funda- 
mental: — X — , rigid wall; • • • x • • •, isotropic wall. 

A more revealing measure of the effectiveness of us- 
ing compliant walls to suppress secondary instabilities is 
to compute the amplitude growth and decay with down- 
stream distance. The amplification of the primary and 
secondary instabilities are governed by 



of waves at a frequency Fr = 53. At this frequency, 
primary amplitudes over the the isotropic wall are sim- 
ilar to those over the rigid wall. This suggests that the 
development of secondary instabilities might also be sim- 
ilar, since secondary disturbances are parametrically de- 
pendent on the basic fiow. For the non-isotropic wall 
results shown in Figure 4, the primary amplitudes are 
suppressed significantly compared to the rigid wall and 
isotropic wall cases. Therefore, the non-isotropic wall 
would likely lead to a very different secondary instability 
development, most probably with reduced amplitudes. 
Again from Figure 4, one might expect greater differ- 
ences in the secondary instability development over both 
compliant walls as the Reynolds numbers increase and 
corresponding frequencies decrease. 



> |- 



n 



r 



m~ 




_I_L 



j-j- 



I I I I I I 



_I_L 



3W3 



<oo sw *w 



700 



Fig. 3 Amplitude growth as a function of 

Reynolds number for the subharmonic mode (B) 
of a 2-D primary wave (A) over a rigid wall at 

Fr= 124, Ao = 0.0044, 5o = 1.86x 10-^ and & = 0.33. 
, theory and (x,o), Kachanov and Levchenko 

[15]. 



■4 



t* dx 



■■4 



—^dx 



(27) 



where Ag, Bg are the initial amplitudes aX Xg, A and B 
are the amplitudes at a downstream distance x, and * 
denotes dimensional quantities. As shown by Herbert^®, 
the theoretical prediction of primary and secondary am- 
plification by eqns. (27) compares well with the exper- 
iments of Kachanov and Levchenko^*^. A similar com- 
parison is shown in Fig. 3, where the theoretical results 
were obtained with the present numerical techniques. 

Before computing similar amplification results over 
the compliant walls, inferences of the secondary insta- 
bility growth may be drawn from primary instability 
results. Figure 4 shows the maximum amplification of 
various frequency primary waves propagating over the 
rigid and compliant walls along with the amplification 




J J L. 



J L 



0.0 



^ ^-^^^ 

hiCTmiUc munbcT, Rr ^ U*"^ 



1.5 



Fig. 4 2-D curves of maximum amplification for 

TSI waves over a , rigid wall; , isotropic 

wall; and , non-isotropic wall and — • — , waves 

of Fr ~ 53. 



To demonstrate these postulations for secondary in- 
stabilities, eqns. (27) are used to compute the amplifi- 
cation of primary and secondary instabilities over the 
rigid, isotropic, and non-isotropic walls for Ag = 0.004, 
Bo = lx 10-^ and h = 0.15, where h = 13/ R x 10^. The 
initial amplitudes are somewhat arbitrary: the present 
values were selected to be close to the experiments of 
Kachanov and Levchenko^*^ for rigid walls. The spanwise 
wavenumber (h) was chosen near the maximum growth 
rate of the secondary instability at the branches of the 
neutral curve for the rigid wall case. Both primary and 
secondary amplifications are shown in Fig. 5. Clearly, 
the results of the secondary instability growth over the 
isotropic and rigid walls are similar, as postulated, while 
the non-isotropic wall significantly suppresses the sec- 
ondary instability growth. 



9 

I 



ft-*' - 




4|i.D2S 






■M CI . n7 :< 
lUymifali:] 

Fig. 5 Amplitude growth as a function of 

Reynolds number for the subharmonic mode (B) 
of a 2-D primary wave (A) at Fr ~ 53, Ag = 0.004, 

Bo = I X 10~®, and h = 0.15 over a , rigid wall; 

, isotropic wall; and , non-isotropic wall. 

The growth of the secondary instability is dependent 
on the parameters of the basic fiow, most probably the 
primary instability amplitude (A). For example. Fig. 6 
shows the amplification of the primary and secondary 
instabilities over the rigid wall with properties as before 
and over the non-isotropic wall with both Ag = 0.004 
and Ag = 0.008. Even by doubling the initial amplitude 
of the primary disturbance, the growth of the secondary 
instability over the non-isotropic wall continues to be 
suppressed and has not exceeded the primary amplitude 
upon crossing the neutral curve. Yet, the doubled initial 
amplitude (Ag = 0.008) results in a significant increase 
in the secondary instability growth compared with the 
lower amplitude (Ag = 0.004) results. Hence, the sup- 
pression of the primary instability amplitude is of utmost 
importance to suppress the onset of the secondary insta- 
bility growth. 




0,0* Q.-DG CDS 0.10 0.12 



14 



Fig. 6 Amplitude growth as a function of 
Reynolds number for the subharmonic mode (B) 
of a 2-D primary wave (A) at Fr ~ 53, Bg = 

1 X 10~®, and b = 0.15 over a , rigid wall 

with Ag = 0.004; — • — , non-isotropic wall with 

Ag = 0.004; and , non-isotropic wall with 

Ag = 0.008. 

.MS I- 



.om - 



.002 - 



.000 




Ifjiimr •iTrnnmh rii P 

Fig. 7 Growth rates of the subharmonic distur- 
bance for 2-D and 3-D primary waves over the 
isotropic wall as a function of spanwise wavenum- 
ber for Rs' = 2240, Fr ~ 22.3, and A = 0.01. , 

<j, = 0«; ■■■,<j> = 10°; and , <j> = 20°. 

Proceeding with investigating the effect of compli- 
ant walls on secondary instabilities, 3-D primary waves 
are introduced and are determined by the specified wave 
angle ((f)). In Fig. 7, subharmonic disturbance growth 
rates over the isotropic compliant wall are shown with 
variation in spanwise wavenumber and primary wave an- 
gle ((/)). A frequency w = 0.05 (Fr ~ 22.3) is selected 
since the 3-D primary wave growth rate is maximized 
at this frequency for the Reynolds number Rs- = 2240. 
Subharmonic growth rates arising from the 2-D primary 
wave ((f) = 0°) are clearly larger than those from the 
3-D waves. As the primary wave angle ((f)) increases, 
the subharmonic growth rates continually become more 
damped. Additionally, for oblique waves ((f) ^ 0°) the 
secondary disturbances no longer travel synchronously 
with the primary wave. This is shown in Fig. 8 by the fre- 



quency shifts that result over the isotropic wall. Similar 
results occur for the non-isotropic wall. It is likely that 
this shift leads to a reduced efficiency of energy transfer 
from the basic flow to the secondary disturbance. This 
frequency shift is as much a result of the 3-D nature of 
the basic flow as it is of the compliant wall influence. Re- 
sults similar to those of Fig. 8 were found by Balachan- 
dar, Streett and Malik^® for the rotating-disk problem 
which also has a 3-D basic flow. 
0.00 I- 



■Q.O? - 



Q.Q* 





' m _ 






• 


X "- . 




\ -. 


- 


\ ' 




\ 


■ 


\ 




. , 1 . . , , 1 . , , , 1 



Q.O 



5.1 



0.2 



9 



e,j 



Fig. 8 Frequency shift of the subharmonic distur- 
bance for 3-D primary waves over the isotropic 
wall as a function of spanwise wavenuniber for 

Rs, = 2240, Fr ~ 22.3, and A = 0.01. ■■■,<j> = 10° 
and , <j> = 20°. 

The secondary growth rate comparison in Fig. 7 is 
misleading as a result of holding the primary amplitude 
(A) flxed. With a variation in wave angle ((f)), corre- 
sponding primary wave amplitudes result as shown by 
Joslin, Morris and Carpenter^®. Then a more realistic 
comparison of secondary growth rates arising from 2-D 
and 3-D primary waves should involve amplitudes suited 
to a given wave angle ((f)). One means to derive these 
amplitudes is through the use of eqns. (27). In Fig. 9, 
a comparison of the subharmonic growth rates over the 
isotropic compliant wall is made for 2-D and 3-D primary 
waves allowing for amplitude differences. Similar results 
are found for the non-isotropic wall as well. As shown, 
the 3-D primary wave does lead to much larger growth 
rates than a 2-D wave for the compliant walls considered. 
This clearly demonstrates that in spite of the loss of syn- 
chronization with the basic flow, 3-D primary waves lead 
to dominant secondary instabilities over compliant walls. 
Again, the amplitude of the primary wave is of utmost 
importance for determining the secondary disturbance 
growth. Yet, compliant walls do lead to a reduction in 
secondary growth rates compared with those for the rigid 
wall. Also, recall that the amplitudes for 3-D primary 
waves used in these calculations were determined by a 
normal mode assumption which leads to a conservative 



estimation of the primary amplitude^®, and a more re- 
alistic lower amplitude would lead to secondary growth 
rates somewhere between the 2-D and 3-D results shown 
in Fig. 9. But, the goal here is to determine the fun- 
damental effect of compliant walls on secondary distur- 
bances. 

Final amplitude calculations using eqns. (27) are 
carried out for the frequency Fr ~ 53. In Fig. 10, sec- 
ondary amplitudes arising from the most amplifled 3- 
D primary wave over a non-isotropic compliant wall are 
compared to the previous results from 2-D primary waves 
over rigid and non-isotropic walls. At this frequency, the 
amplitude difference between the 2-D and 3-D primary 
instabilites is small, yet the secondary disturbance re- 
sponds notably. This is an indication that small changes 
in the primary instability, however slight, have a mount- 
ing effect on the rapidly developing secondary instability. 



,0490 



i.onss 



.oooa 




a.o 



a. 2 



i>.i 



0,4 



Fig. 9 Growth rates of the subharmonic distur- 
bance for 2-D and 3-D primary waves over the 
isotropic wall as a function of spanwise wavenuni- 
ber for Rs' = 1760 and Fr ~ 30.2 for , <j> = 0" 

with A = 0.010; • • • , ,?i = 45° with A = 0.031. 

I |- 



5*"-^ - 



1 

^ 




.^^ 



fr 



p.na n.iH 



o.oa 



O.OJt o.ia O.IS 



D.I4 



Fig. 10 Amplitude growth as a function of 
Reynolds number for the subharmonic mode (B) 
of a 2-D and 3-D primary wave (A) at Fr ~ 53, 

Bo = lx 10-^ Ao = 0.004, and h = 0.15 over a , 

rigid wall — • — , 2-D non-isotropic wall; and • • •, 
3-D non-isotropic wall. 



6. Summary 

In earlier studies^®'^®'^^, it was shown that 3-D pri- 
mary instabilities theoretically dominate transition over 
the compliant walls considered, yet transition delays 
were found compared to the rigid wall. The present pa- 
per has further extended the understanding of the effect 
of compliant walls on transition mechanisms in bound- 
ary layers. Namely, the effect compliant walls have on 
secondary instabilities has been investigated. It has been 
shown that the use of compliant walls can lead to reduced 
growth rates and amplification of secondary instabili- 
ties. From both the earlier 3-D primary results combined 
with the present secondary findings, it should be empha- 
sized that the physical nature and make-up of the mech- 
anisms in transition are not altered by the control device 
(i.e. compliant wall). Rather, only the response of that 
mechanism is changed. This fact is of particular impor- 
tance for designing Laminar Flow Control (LFC) studies. 
As an example, the behavior of a secondary instability 
growth with variation in primary amplitude is well doc- 
umented by Herbert. As the primary amplitudes are 
reduced, the excitement of the secondary instability is 
delayed. Thus, active or passive devices which suppress 
primary instability growth should lead to corresponding 
suppression and delay of succeeding instabilities. This 
has been demonstrated above with the compliant wall. 
The twofold major discovery and demonstration of the 
present investigation is: (1) the use of passive devices, 
such as compliant walls, lead to significant reductions in 
the secondary instability growth rates and amplification; 
(2) suppressing the primary growth rates and subsequent 
amplification enable delays in the growth of the explosive 
secondary instability mechanism. 

7. Acknowledgments 

Support for this project was supplied by the Naval 
Sea Systems Command and the Applied Research Labo- 
ratory Exploratory and Foundational Research Program 
under NAVSEA N0039-88-C-0051 at the Department of 
Aerospace Engineering, The Pennsylvania State Univer- 
sity. A grant for computational support was provided 
by the National Science Foundation with the Pittsburgh 
Supercomputing Center. 

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10 



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11